simple meromorphic functions are algebraic

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Bull Braz Math Soc, New Series 44(2), 309-319© 2013, Sociedade Brasileira de Matemática

Simple meromorphic functions are algebraic

Guy Casale

Abstract. We exhibit a class of meromorphic functions in two variables conjugated toalgebraic functions using the geometry of the foliation by level curves.

Keywords: differential equations, dicritical singularities.

Mathematical subject classification: 32S65.

We are interested in the following problem:

Let F be a germ of holomorphic foliation on (C2, 0). Does thereexist an algebraic surface S and a point p ∈ S such that F is thegerm at p of an algebraic foliation on S?

Such germs of foliation will be said to be algebraizable. In [6], Y. Genzmerand L. Teyssier proved the existence of a non algebraizable germ of saddle-nodefoliation. After them the problem splits in two parts:

Problem. Give an example of non algebraizable germ of singularity.

Problem. Identify algebraizable singularities.

We give an answer to the second problem in a very particular case followingfirst pages of [4] (see also [9]).

Since Mather and Yau [13], it is known that a germ of holomorphic functionwith isolated singularity is finitely determined thus algebraizable. Such resultwas extended by Cerveau and Mattei [3] to germs of meromorphic functions. Aconsequence of the result of this paper is to get an example of a germ of algebraicmeromorphic function which is not finitely determined.

Received 8 October 2012.

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310 GUY CASALE

This paper is concerned with holomorphic foliationsof (C2, 0) with a dicriticalsingularity at 0. Such germs of foliations have infinitely many analytic invariantcurves going through 0. Basic examples of dicritical foliations are foliationsgiven by level sets of a meromorphic function on (C2, 0) having indeterminacypoint at 0. Among all these singularities with a meromorphic first integral, weare interested in the simplest ones i.e. those smooth after one blow-up and suchthat a unique leaf is tangent to the exceptional divisor with tangency order one.These singularities will be called simple singularities.

Example (Basic example: the cusp). The dicritical cusp is given by the ratio-nal function y2−x3

x2 :

After one blow-up one gets the function t2 − x defining the foliation on thechart t = y/x .

Example (Susuki’s example [16, 3]). This is the germ of singularity given by

(y3 + y2 − xy)dx − (2xy2 + xy − x 2)dy

It is topologically equivalent to the previous example. The topological closureof a leaf contains 0 and is analytic. But its ‘less transcendental’ first integral is

x

ye

y(y+1)x .

It does not admit meromorphic first integral in any neighborhood of 0.

Theorem 1. If F is a simple dicritical foliation of (C2, 0) with a meromorphicfirst integral then there exist an algebraic surface S, a rational function H onS and a point p ∈ S such that F is biholomorphic to the foliation given levelcurves of H in a neighborhood of p.

Corollary 2. If F is a germ of meromorphic function on (C2, 0) with a simpletype indeterminacy point at 0 then it is the germ of a rational function on asurface.

Proof of the Corollary. After a change of coordinates given by Theorem 1,the level curves of F coincide with those of a rational function H on an algebraicsurface at some point 0 such that F = f (H ) for some function f defined onsome ramified covering of P1. On the exceptional divisor obtained by blowing-up 0, F and H are rational functions so f is algebraic and F is a germ of rationalfunction on a surface. �

Analytic invariant of simple singularities and basic constructions are given in§1. In §2 a proof of Theorem 1 is given. Last section contains some comments.

Bull Braz Math Soc, Vol. 44, N. 2, 2013

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SIMPLE MEROMORPHIC FUNCTIONS ARE ALGEBRAIC 311

1 The analytic invariant of simple singularities

Following M. Klughertz [10], the description of a simple singularity up toanalytic change of coordinates is done by a germ of involution at a point ofP1 up to the action of PGL2(C). Let F be a germ of simple dicritical foliationof (C2, 0). The blow-up of (C2, 0) is denoted by M , the exceptional divisor byE and the strict transform ofF byF−1. BecauseF−1 is smooth and has a uniqueleaf tangent to E at order one at some point p, leaves passing through a point inthe neighborhood of p on E must cut E in another point. This gives us a germof involution ι on the projective line. This is the holonomy of F on E .

The germ of involution of two analytically conjugated foliations are the sameup to conjugation by an element of PGL2(C). Conversally if two germs offoliations with simple singularities have the same involution then they are analyt-ically conjugated. To complete M. Klughertz’ theorems, some finite determinacyproperties and normal forms are obtained in [1, 15].

Let (M, E) be a pair of analytic surface and smooth rational curve E ⊂ Mwith self-intersection E · E = k. A neighborhood of E in M will still be denotedby M and called a k-neighborhood of P1.

Lemma 3. Assume k ≤ 0. Any germ of involution on P1 can be realized as theholonomy of a smooth holomorphic foliation of a k-neighborhood of P1. Twosuch realizations of the same involution are biholomorphic. For a involution ι,any such realization is called a k-realization of ι and denoted by Fk(ι).

Proof. The proof is an illustration of the gluing trick of [12]. Let ι be suchan involution on a disc Dt (r) with coordinate t and radius r and ϕ be de-fined by ϕ(t) = t − ι(t). The involution ι is the holonomy of the foliation ofDt (r) × Dx(r ′) given by level curves of ϕ(t)2 − x . Let r ′ be small enough thenU = {t ∈ Dt (r) | |ϕ(t)2| ≤ r ′} ⊂⊂ Dt (r). Leaves of the foliation by levelcurves of ϕ(t)2 − x on (Dt − U ) × Dx intersect Dt − U in a single point. Forthis reason the function

√ϕ(t)2 − x is well defined on (Dt − U ) × Dx .

Let M be the manifold obtained by gluing Dt × Dx with (P1 − U )u × Dy by

u = ϕ−1(√

ϕ(t)2 − x)

and y = xt−k.

The foliation given by d(ϕ(t)2 − x) = 0 on the first chart and du = 0 on thesecond is well defined, transverse to E the divisor given by x = 0 and y = 0but in a single point t = 0. The holonomy is ι by construction. When k ≤ 0,Grauert’s theorems [5, 8] give unicity of the germ of neighborhood built. �


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Remark 4. The contraction of the exceptional divisor in the (−1)-realization ofι gives a dicritical foliation of (C2, 0) with invariant ι.

Remark 5. By blowing-up a point of the projective line which is not the fixedpoint of the involution ι, Fk(ι) is transformed in Fk−1(ι).

Remark 6. If one glues a node d(xy) = 0 to Fk−1(ι) in such a way that aseparatrix of the node is glued with a leaf of Fk−1(ι) in a rational curve ofself-intersection (−1) then the contraction of this leaf gives Fk(ι).

Remark 7. The involution ι has a rational first integral, i.e. R ∈ C(P1) suchthat R ◦ ι = R, if and only if Fk(ι) has a meromorphic first integral.

2 Proof of Theorem 1

The strategy is to normalized a 0-realization of the involution of a dicriticalsingularity with a meromorphic first integral, and to prove that the meromorphicfirst integral of the dicritical foliation is transformed in an algebraic first integralof the 0-realization. By Remarks 4-7 this is enough to prove the Theorem. LetF−1(ι) be a blow-up of a simple dicritical singularity with a meromorphic firstintegral and F0 be a 0-realization of the involution. It is defined on D × P

1

where D is a small disc around 0 in C. Let y be a coordinate on P1 and x be acoordinate on D. Such a foliation is given by a differential equation

dy

dx= P3(y)

P1(y)

where Pi ∈ C{x}[y] is of degree i in y. By changing y, we can rectify threetrajectories passing through p1, p2 and p3 at x = 0 on straight lines y = ∞,y = 1 and y = −1 and the equation becomes

dy

dx= h(x)

y2 − 1

y − λ(x).

Because the foliation is not singular, h(0) �= 0. Furthermore p1, p2 and p3 canbe choosen such that λ(0) �= 0. Then the trajectory passing through 0 ∈ P1 atx = 0 is the graph of a biholomorphism and can be used as a new coordinate onthe disc. The foliationF0 in these new coordinates is given by the equation

x 2 − 1

x − λ(x)

dy

dx= y2 − 1

y − λ(x)


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for a germ of analytic function λ (different from the previous one) whose graphis the locus of verticality of leaves.

Let H (x , y) ∈ C({x})(y) be a meromorphic first integral of F0 and d be itsdegree in y. Let R(d) be the variety of degree d rational functions on P1. Thisis a dimension 2d +1 algebraic variety. By restriction to vertical fibers, H givesa germ of curve γ : D → R(d) defined by γ (x) = H |fiber above x .

To prove that H is an algebraic function one has to prove that γ is included inan algebraic curve in C×R(d). To do so one will prove that

• γ is an integral curve of a rational vector fields V on an algebraic ramifiedcovering of C×R(d),

• V has 2d + 1 algebraically independent rational first integrals. Thenintegral curves of V are algebraic.

Equations of the deformation of a rational function in the way described bythe drawing ofF0 can be written down explicitely. Let us parameterizeR(d) bythe zeros and the poles of fractions:

R(y) = ��(y − ai)

�(y − bi).

Let R̃ be the ramified covering of R(d) where all the critical points are well-defined i.e. R̃ is the Galoisian covering of R(d) built from the covering ofR(d) defined by the equation

∑ 1p−ai

− ∑ 1p−bi

= 0 in R(d) × C with p thecoordinate on C.

The vector field

V = x 2 − 1

x − λ

∂

∂x+

∑ a2i − 1

ai − λ

∂

∂ai+

∑ b2i − 1

bi − λ

∂

∂bi

where λ is the coodinate function in C(R̃) of a critical point, describes partic-ular deformation of rational functions along a parameter x . If ai (x), bi(x) and�(x) parameterize an analytic integral curve of this vector field then the mero-morphic function

H (x , y) = �(x)�(y − ai(x))

�(y − bi(x))

is a first integral ofx 2 − 1

x − λ(x)

dy

dx= y2 − 1

y − λ(x)

where λ(x) is the restriction of λ to the integral curve.


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Thus 0-realization of involutions with meromorphic first integrals of degreed in y are exactly described by trajectories of V on C × R̃. One will denoteby Hγ the meromorphic function in C{x}(y) given by an integral curve γ of V .Remark that trajectories realizing Hγ and g(Hγ ) for g ∈ PSL2(C) are differentbut describe the same 0-realization of the same involution.

The 2n + 1 independent algebraic first integrals of V are buit in the follow-ing way.

Let us consider the critical values of rational functions as 2n − 2 rationalfunctions on R̃:

c1, . . . , c2d−2.

The restrictions of these functions on an integral curve γ give the criticalvalues of Hγ with respect to y as functions of x . One can assume that c2d−2(γ ) =λ. The 2n − 3 remaining functions give the values of Hγ on leaves where Hγ

ramifies so they are constant on integral curves of V .The values of Hγ on the leaves y = ∞, y = 1, y = −1 and y = x are

constant hence the evaluations function e∞(x , R) = R(∞), e1(x , R) = R(1),e−1(x , R) = R(−1) and ex(x , R) = R(x) are 4 rational functions on D × R̃which are constant on integral curves of V .

In coordinates these functions are:

• ck(...ai...bi...�) = ��i (pk − ai )

�i (pk − bi )for all pk �= λ such that

∑ 1pk−ai

−∑ 1pk−bi

= 0,

• e∞(...ai...bi...�) = �,

• e1(...ai...bi...�) = ��i (1 − ai)

�i (1 − bi),

• e−1(...ai...bi...�) = ��i (−1 − ai)

�i (−1 − bi),

• ex(...ai...bi...�) = ��i(x − ai)

�i(x − bi),

These 2n + 1 funtions are algebraically independent because there is onlya finite number of rational maps from P1 to P1 with fixed critical valuesc1, . . . , c2d−2 and fixed values at −1, 1, ∞. These maps are given by thechoice of the monodromies around critical values in the group of permutationof {1, 2, . . . , d}. Thus the functions c1, . . . , c2d−2, e1, e−1, e∞ are independent


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on R̃ and so are c1, . . . , c2d−3, e1, e−1, e∞ on D × R̃. These functions are x -independent so are independent of ex . This proves the functional independenceof 2n + 1 rational first integrals of V a vector field on a dimension 2n + 2 alge-braic variety. The Lemma proved at the end of the section gives the algebraicityof integral curves of V .

Let S be an algebraic curve such that γ is defined on. The function H extendsto a rational function on S ×P1. By blowing-up a point on 0 ×P1 and blowing-down the strict transform of 0×P1 one gets the dicritical singularitywith rationalfirst integral and ι as invariant.

This proves that a simple dicritical foliation given by a meromorphic functioncan be extended (in good coordinates) in a algebraic foliation with a rationalfirst integral. �

The following lemma is a well-known fact but it is not easy to find a reference.

Lemma 8. If a vector field V on a dimension n algebraic variety has n − 1rational first integrals functionally independent thenall trajectories are algebraiccurves.

Proof. For rational functions, H1, . . . , Hk, functional independence (dH1 ∧. . .∧ dHk �= 0) and algebraic independance coincide.

LetK be the field of rational first integrals of V . It is a transcendence degreen − 1 field. Let H1, . . . Hn−1 be a transcendence basis. Then, out of the depen-dency set {dH1 ∧ . . . ∧ dHn−1 = 0} and the indeterminacy sets of H ’s, integralcurves are algebraic.

Let S be a V -invariant irreducible hypersurface on which dH1 ∧ . . .∧dHn−1 = 0. One wants to find n − 2 elements of K whose restrictions onS are algebraically independent.

Let x ∈ S be a generic point and f a local holomorphic ireducible equation ofS at x . One defines d : K∗ → Z by H = f d(H ) P

Q with P and Q holomorphicfunction not identically zero on S. There exists F ∈ K such that d(K∗) =d(F)Z.

Assume one gets F1, . . . , Fk−1 whose restrictions to S are functionnally inde-pendent. For H ∈ K one defined �(H ) such that dH ∧ dF1 ∧ . . . ∧ dFk−1 =F�(H )ω with ω|S a rational form in the neighborhood of S non zero on S. Ifsuch a � does not exist, it is defined to by −∞. If k − 1 < n − 2 there existsH ∈ K such that F, F1, . . . , Fk−1 and H are functionnally independent andthese functions are holomorphic at x then �(K) ∩ N is not empty.

Let Fk ∈ K such that �(Fk) is minimal in �(K) ∩ N. If �(Fk) = 0 then thelemma is proved.


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If �(Fk) > 0 then let R(F1|S, . . . , Fk|S) be the minimal polynomial of Fk|Sover C(F1|S, . . .Fk−1|S) and considere F̃k = R(F1,...,Fk)

F . This function is holo-morphic at x because d(F) si smaller than d(R(F1, . . . , Fk)). One has

d F̃k ∧ dF1 ∧ . . . ∧ dFk−1 =(

1

F

∂ R

∂FkdFk + R

F2dF

)∧ dF1 ∧ . . . ∧ dFk−1.

Since R is the minimal polynomial of Fk |S ∂ R∂Fk

can not vanish on S if R is not

zero. One has 0 ≤ �(F̃k) < �(Fk) which is a condradiction then R = 0 and thelemma is proved. �

3 Comments

3.1 Cerveau-Mattei finite determinacy theorem

For special kind of functions there exists an algebraization result proved by D.Cerveau and J.-F. Mattei in [3]. Let f1, . . . , f p be germs of irreducible holo-morphic functions in (Cn, 0), α1, . . . , αp be complex numbers. Considere themultivalued function

H = f α11 . . . f

αpp .

Let ω be the form f1 . . . f p∑

αid fifi

and X be the hypersurface f1 . . . f p = 0.

Definition 9. The small critical locus of H is a subset C′(H ) of Zero(ω) ∪ Xof point p such that if p ∈ X germs at p of f ’s are reducible or (X, p) is not anormal crossing germ of hypersurface.

The function H is said to be finitely determined at order k if for everyg1, . . .gp with jk( fi − gi) = 0 there is a diffeomorphism u of (Cn, 0) suchthat

H ◦ u = gαp1 . . .g

αpp

Theorem 10 (Cerveau-Mattei [3] Théorème 4.2. p. 163). A function H asabove is finitely determined if and only if C′(H ) ⊂ {0}.

In our situation, n = 2, αi ∈ Z, finite determinacy implies conjugation to arational function with same ‘order k jet’ at 0. There is a rational function whichis not finitely determined i.e. such that a meromorphic function with the sameorder k jets of numerator and denominator is not conjugated to the previousrational functions. Such an example can be build using previous description ofsimple dicritical foliations.


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Let ι be the germ of involution at 0 ∈ P1 defined by

h ◦ ι = h

for some rational function h with a double zero at 0. For instance h(t) =t2((t − 1)2 + 1). The (−1)-realization of ι has a meromorphic first integralH extending h to the (−1)-neighborhood of P1 followingF−1(ι). By blowing-down this function, one gets a germ of meromorphic function on (C2, 0) withsimple indeterminacy point.

By Cerveau-Mattei theorem such a function is finitely determined if and onlyif its small critical locus is contained in {0}. But the critical set of H is con-tained in Zero(ω) and coincides with it out of the zero and polar locus of H .If the small critical locus is included in {0} then Zero(ω) is included in thezero and polar locus of H . This is not the case for the function built from h.This H is f0 f1+i f1−i

f 4∞where f p is an equation of the leaves having slope p at 0.

Its critical set is given by three leaves with slopes 3/4 + i√

5/4, 3/4 − i√

5/4and ∞ at 0.

To prove that H is algebraizable by means of this finite determinacy theorem,one only needs to find a rational function f such that f ◦H is finitely determinate.Critical values of f ◦ H are critical values of f and the image by f of thoseof H . If f ◦ H satisfies the hypothesis of the finite determinacy theorem, thesecritical values must be 0 and ∞. Such a f would be a (non ramified) coveringof P1 − {0, ∞} by P1 − D where D is a finite set containing critical values ofH . If #D ≥ 3 this is not possible. It is the case for our special H coming fromh where

D ⊃ {0, h(3/4 + i

√5/4), h(3/4 − i

√5/4), ∞}

and for this reason there is no f such that f ◦ H is finitely determined.Nevertheless by our theorem, this function is a germ of algebraic function in

suitable coordinates.

3.2 The λ is not unique

If one fixes the involution ι and three points on P1 then the function λ givenby the normalisation of a 0-realization F0 is unique. So λ is an invariant of themarked involution (ι, p−1, p1, p∞) on P1. The involution ι itself is an invariant.

If four differents leaves are normalized on y = ∞, y = −1, y = 1 andy = x one gets a new function λ∗. Let us explain this change of λ. Letu1(x), u−1(x), u∞(x) be three solutions of F0 and u0 be the solution such


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that the crossratio of (u∞(0), u−1(0), u0(0), u1(0)) equals the cross-ration of(∞, −1, 0, 1). The change of variables⎧⎪⎪⎪⎨

⎪⎪⎪⎩x∗ = u1 − u∞

u0 − u∞+ u0 − u1

u−1 − u∞

y∗ = u1 − u∞y − u∞

+ y − u1

u−1 − u∞

rectifies the u’s on the four lines y∗ = ∞, y∗ = 1, y∗ = −1 and y∗ = x∗and gives another normalisation of the 0-realization of ι. The involution of thisrealisation is ι∗ defined by

ι∗(

u1(0) − u∞(0)

y − u∞(0)+ y − u1(0)

u−1(0) − u∞(0)

)= u1(0) − u∞(0)

ι(y) − u∞(0)+ ι(y) − u1(0)

u−1(0) − u∞(0).

If λ∗ is the function given by the verticality locus of this new diffferential equationthen one gets

λ∗(

u1 − u∞u0 − u∞

+ u0 − u1

u−1 − u∞

)= u1 − u∞

λ − u∞+ λ − u1

u−1 − u∞.

References

[1] G. Calsamiglia-Mendlewicz. Finite determinacy of dicritical singularities in(C2, 0). Ann. Inst. Fourier, 57 (2007), 673–691.

[2] C. Camacho and H. Movasati. Neighborhoods of analytic varieties, volume 35 ofMonografas del Instituto de Matematica y Ciencias Afines IMCA, Lima (2003).

[3] D. Cerveau and J.-F. Mattei. Formes intégrables holomorphes singulières. Asté-risque, 97 (1982), 193 pp.

[4] J. Drach. L’équation différentielle de la balistique extérieur et son intégrationpar quadratures. Ann. Sci. École Normale Sup., 37 (1920), 1–94.

[5] W. Fischer and H. Grauert. Lokal-triviale familien kompakter komplexer mannig-faltigkeiten. Nach Akad. Wiss. Gottingen Math.-Phys. KL. II, (1965), 89–94.

[6] Y. Genzmer and L. Teyssier. Existence of non-algebraic singularities of differen-tial equation. J. Differential Equations, 248(5) (2010), 1256–1267.

[7] X. Gomez-Mont. Integrals for holomorphic foliationswith singularities having allleaves compact. Ann. Inst. Fourier, 39(2) (1989), 451–458.

[8] H. Grauert. Über Modifikationen und exzeptionelle analytische Mengen. Math.Ann., 146 (1962), 331–368.

[9] G. Heilbronn. Intégration des équations différentielles ordinaires par la méthodede Drach. Mémoire. Sci. Math., no. 133 Gauthier-Villars, Paris (1956), 103 pp.


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[10] M. Klughertz. Feuilletages holomorphes à singularité isolée ayant une infinité decourbes intégrales. Ph.D Thesis Université Toulouse III (1988).

[11] M. Klughertz. Existence d’une intégrale première méromorphe pour des germesde feuilletages à feuilles fermées du plan complexe. Topology, 31 (1992), 255–269.

[12] F. Loray. Analytic normal forms for non degenerate singularities of planar vectorfields, preprint CRM Barcelona, 545 (2003), http://www.crm.es/Publications/03/pr545.pdf.

[13] J.N. Mather and S.T. Yau. Classification of isolated hypersurface singularities bytheir moduli algebras. Invent. Math., 69 (1982), 243–251.

[14] R. Meziani and P. Sad. Singularités nilpotentes et intégrales premières, preprintIMPA, A415 (2005), http://www.preprint.impa.br/Shadows/SERIE_A/2005/415.html.

[15] L. Ortiz-Bobadilla, E. Rosales-Gonzalez and S.M. Voronin. Rigidity theoremsfor generic holomorphic germs of dicritic foliations and vector fields in (C2, 0).Moscow Math. J., 5 (2005), 171–206.

[16] M. Susuki. Sur les intégrales premières de certains feuilletages analytiques com-plexes. Fonct. de plusieurs variables complexes, sémin. Francois Norguet, Lect.Notes Math., 670 (1978), 53–79.

Guy CasaleIRMAR-UMR CNRS 6625Université de Rennes 1F-35042 Rennnes cedexFRANCE

E-mail: [email protected]


simple meromorphic functions are algebraic

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